As an integrable background let us recall the relation between the FL equation and the KN hierarchy.
The KN spectral problem reads [1]}, [2]}
\( \begin{pmatrix} \varphi _1\\ \varphi _2\end{pmatrix}_x=\begin{pmatrix}\frac{i}{2}\lambda ^2 & \lambda q \\ \lambda r &-\frac{i}{2}\lambda ^2 \end{pmatrix}\begin{pmatrix} \varphi _1\\ \varphi _2\end{pmatrix},\) 
